D Thermodynamics of ideal gases An ideal gas is a nice “laboratory” for understanding the thermodynamics of a fluid with a non-trivial equation of state. In this section we shall recapitulate the conventional thermodynamics of an ideal gas with constant heat capacity. For more extensive treatments, see for example [67, 66]. D.1 Internal energy In section 4.1 we analyzed Bernoulli’s model of a gas consisting of essentially 1 2 non-interacting point-like molecules, and found the pressure p = 3 ½ v where v is the root-mean-square average molecular speed. Using the ideal gas law (4-26) the total molecular kinetic energy contained in an amount M = ½V of the gas becomes, 1 3 3 Mv2 = pV = nRT ; (D-1) 2 2 2 where n = M=Mmol is the number of moles in the gas. The derivation in section 4.1 shows that the factor 3 stems from the three independent translational degrees of freedom available to point-like molecules. The above formula thus expresses 1 that in a mole of a gas there is an internal kinetic energy 2 RT associated with each translational degree of freedom of the point-like molecules. Whereas monatomic gases like argon have spherical molecules and thus only the three translational degrees of freedom, diatomic gases like nitrogen and oxy- Copyright °c 1998{2004, Benny Lautrup Revision 7.7, January 22, 2004 662 D. THERMODYNAMICS OF IDEAL GASES gen have stick-like molecules with two extra rotational degrees of freedom or- thogonally to the bridge connecting the atoms, and multiatomic gases like carbon dioxide and methane have the three extra rotational degrees of freedom. Accord- ing to the equipartition theorem of statistical mechanics these degrees of freedom 1 will also carry a kinetic energy 2 RT per mole. Molecules also possess vibrational degrees of freedom that may become excited at high temperatures, but we shall disregard them here. The internal energy of n moles of an ideal gas is defined to be, k U = nRT ; (D-2) 2 where k is the number of molecular degrees of freedom. A general result of thermodynamics (Helmholtz’ theorem [67, p. 154]) guarantees that for an ideal gas U cannot depend on the volume but only on the temperature. Physically a gas may dissociate or even ionize when heated, and thereby change its value of k, but we shall for simplicity assume that k is in fact constant with k = 3 for monatomic, k = 5 for diatomic, and k = 6 for multiatomic gases. For mixtures of gases the number of degrees of freedom is the molar average of the degrees of freedom of the pure components (see problem D.1). D.2 Heat capacity Suppose that we raise the temperature of the gas by ±T without changing its volume. Since no work is performed, and since energy is conserved, the necessary amount of heat is ±Q = ±U = CV ±T where the constant, k C = nR ; (D-3) V 2 is naturally called the heat capacity at constant volume. If instead the pressure of the gas is kept constant while the temperature is raised by ±T , we must also take into account that the volume expands by a certain amount ±V and thereby performs work on the surroundings. The necessary amount of heat is now larger by this work, ±Q = ±U + p±V . Using the ideal gas law (4-26) we have for constant pressure p±V = ±(pV ) = nR±T . Consequently, the amount of heat which must be added per unit of increase in temperature at constant pressure is Cp = CV + nR ; (D-4) called the heat capacity at constant pressure. It is always larger than CV because it includes the work of expansion. Copyright °c 1998{2004, Benny Lautrup Revision 7.7, January 22, 2004 D.3. ENTROPY 663 The adiabatic index The dimensionless ratio of the heat capacities, C 2 γ = p = 1 + ; (D-5) CV k is for reasons that will become clear in the following called the adiabatic index. It is customary to express the heat capacities in terms of γ rather than k, 1 γ C = nR ; C = nR : (D-6) V γ ¡ 1 p γ ¡ 1 Given the adiabatic index, all thermodynamic quantities for n moles of an ideal gas are completely determined. The value of the adiabatic index is γ = 5=3 for monatomic gases, γ = 7=5 for diatomic gases, and γ = 4=3 for multiatomic gases. D.3 Entropy When neither the volume nor the pressure are kept constant, the heat that must be added to the system in an infinitesimal process is, ±V ±Q = ±U + p±V = C ±T + nRT : (D-7) V V It is a mathematical fact that there exists no function, Q(T;V ), which has this expression as differential (see problem D.2). It may on the other hand be directly verified (by insertion) that ±Q ±T ±V ±S = = C + nR ; (D-8) T V T V can be integrated to yield a function, S = CV log T + nR log V + const ; (D-9) called the entropy of the amount of ideal gas. Being an integral the entropy is only defined up to an arbitrary constant. The entropy of the gas is, like its energy, an abstract quantity which cannot be directly measured. But since both quantities depend on the measurable thermodynamic quantities, ½, p, and T , that characterize the state of the gas, we can calculate the value of energy and entropy in any state. But why bother to do so at all? The two fundamental laws of thermodynamics The reason is that the two fundamental laws of thermodynamics are formulated in terms of the energy and the entropy. Both laws concern processes that may Copyright °c 1998{2004, Benny Lautrup Revision 7.7, January 22, 2004 664 D. THERMODYNAMICS OF IDEAL GASES take place in an isolated system which is not allowed to exchange heat with or perform work on the environment. The First Law states that the energy is unchanged under any process in an V isolated system. This implies that the energy of an open system can only change by exchange of heat or work with the environment. We actually used this law implicitly in deriving the heat capacities and the entropy. ....... The Second Law states that the entropy cannot decrease. In the real world, ......... ..................... V0 ................... the entropy of an isolated system must in fact grow. Only if all the processes .......... ........ taking place in the system are completely reversible at all times, will the entropy stay constant. Reversibility is an ideal which can only be approached by very slow quasistatic processes, consisting of infinitely many infinitesimal reversible steps. Essentially all real-world processes are irreversible to some degree. A compartment of volume V0 inside an isolated box of volume V . Initially, Example D.3.1 (Joule’s expansion experiment): An isolated box of vol- the compartment contains ume V contains an ideal gas in a walled-off compartment of volume V0. When the an ideal gas with vacuum wall is opened, the gas expands into vacuum and fills the full volume V . The box in the remainder of the is completely isolated from the environment, and since the internal energy only de- box. When the wall breaks, the gas expands by itself pends on the temperature, it follows from the First Law that the temperature must to fill the whole box. The be the same before and after the event. The change in entropy then becomes reverse process would entail a decrease in entropy and ∆S = (CV log T + nR log V ) ¡ (CV log T + nR log V0) = nR log(V=V0) never happens by itself. which is evidently positive (because V=V0 > 1). This result agrees with the Second James Prescott Joule (1818- Law, which thus appears to be unnecessary. 1889). English physicist. The strength of the Second Law becomes apparent when we ask the question Gifted experimenter who as of whether the air in the box could ever — perhaps by an unknown process to be the first demonstrated the discovered in the far future — by itself enter the compartment of volume V0, leaving equivalence of mechanical vacuum in the box around. Since such an event would entail a negative change in work and heat, a necessary step on the road to the First entropy which is forbidden by the Second Law, it never happens. Law. Demonstrated (in con- tinuation of earlier experi- ments by Gay-Lussac) that Isentropic processes the irreversible expansion of a gas into vacuum does not Any process in an open system which does not exchange heat with the environ- change its temperature. ment is said to be adiabatic. If the process is furthermore reversible, it follows that ±Q = 0 in each infinitesimal step, so that the ±S = ±Q=T = 0. The entropy (D-9) must in other words stay constant in any reversible, adiabatic process. Such a process is for this reason called isentropic. By means of the adiabatic index (D-5) we may write the entropy (D-9) as, ¡ γ¡1¢ S = CV log TV + const : (D-10) From this it follows that TV γ¡1 = const ; (D-11) for any isentropic process in an ideal gas. Using the ideal gas law to eliminate V » T=p, this may be written equivalently as, T γ p1¡γ = const : (D-12) Copyright °c 1998{2004, Benny Lautrup Revision 7.7, January 22, 2004 D.3. ENTROPY 665 Eliminating instead T » pV , the isentropic condition takes its most common form, p V γ = const : (D-13) Notice that the constants are different in these three equations.